Many circuits of practical importance operate in strongly nonlinear regimes under multi-tone (i.e., with several independent time-scales or frequencies) excitations. Familiar examples are switched-capacitor circuits and switching power converters; more recently, RF designs for portable communications have started incorporating new topologies from baseband IC circuits, which often have strongly nonlinear sections.
In typical RF applications, the multi-tone inputs are quasi-periodic (see K. S. Kundert, J. K. White, and A. Sangiovanni-Vincentelli, Steady-State Methods For Simulating Analog And Microwave Circuits, Kluwer Academic Publishers, 1990) or envelope-modulated (see D. Sharrit. "New Method of Analysis of Communication Systems", MTTS96 WMFA: Nonlinear CAD Workshop, June 1996), with timescales of different tones separated by orders of magnitude.
Two classes of methods exist for computing a periodic or quasi-periodic steady state solution of a circuit: shooting/FDTD and harmonic balance.
Shooting and FDTD, both time-domain methods, are closely related. Shooting (see T. J. Aprille and T. N. Trick, "Steady-state analysis of nonlinear circuits with periodic inputs," Proc. IEEE, 60(1):108-114, January 1972; and S. Skelboe, Computation of the periodic steady-state response of nonlinear networks by extrapolation methods. IEEE Trans. Ckts. Syst., CAS-27(3):161-175, March 1980) is based on finding an initial state for a circuit's differential equations such that the circuit returns to the same state after one period of the input. From a guess for the initial state, the circuit's final state after one period is determined by a standard transient analysis. If the final state is not equal to the initial one, the guess is refined using a newton or relaxation algorithm around the circuit's state transition function. FDTD methods solve for the circuit's values over an input period simultaneously rather than sequentially as in shooting. Recently, iterative linear algebra techniques have been applied (see R. Telichevesky, K. Kundert, and J. White, "Efficient Steady-State Analysis based on Matrix-Free Krylov Subspace Methods," Proc. IEEE DAC, pages 480-484, 1995) to these methods, making them efficient for large circuits.
These time-domain methods can analyze strongly nonlinear circuits easily. A significant weakness, however, has been their inability to handle widely-separated multi-tone problems efficiently. The usual approach has been to treat a multi-tone problem as a single-tone one of period equal to that of the slowest tone. This leads to simulations of thousands or millions of periods of the fast tones, which can be very inefficient.
Harmonic balance (HB) methods address the problem in the frequency domain (see R. J. Gilmore and M. B. Steer, "Nonlinear Circuit Analysis using the Method of Harmonic Balance--a review of the art. Part 1. Introductory Concepts," Int. J on Microwave and Millimeter Wave CAE, 1(1), 1991). All waveforms in the circuit are expressed in truncated multi-tone Fourier series; the circuit's differential equations are recast as a set of algebraic equations in the series coefficients. These equations are solved numerically with a Newton-Raphson or relaxation scheme. Traditional algorithms for HB were limited to analyzing relatively small circuits; this problem was recently overcome by exploiting factored-matrix forms for the HB jacobian matrix and using interative linear solvers (see R. C. Melville, P. Feldmann, and J. Roychowdhury, "Efficient Multi-tone Distortion Analysis of Analog Integrated Circuits," Proc. IEEE CICC, May 1995).
Harmonic balance is the method of choice for multi-tone analysis if a circuit has the following features: (1) all waveforms can be represented accurately by Fourier series with relatively few terms; and (2) nonlinearities are mild to moderate. Since traditional microwave and RF circuits have these properties, 1B has been used extensively for these applications. However, recent RF designs for portable communications incorporate new topologies imported from baseband IC circuits, which often have sections that are strongly nonlinear. Strongly nonlinear circuits generate waveforms that are not well-represented by short Fourier series, hence harmonic balance is inefficient for such applications. However, if the inputs themselves are sharp waveforms that do not admit of a compact representation in a Fourier basis, HB can be ineffective even for mildly nonlinear circuits switched-capacitor filters, switching mixers).
Another weakness relates to the structure of the HB jacobian matrix, which is block-diagonally dominant if the nonlinearities are weak, but ceases to be so if the nonlinearities are strong. Recent HB algorithms for large circuits rely on preconditioners for solving the jacobian iteratively; existing preconditioners often become ineffective as diagonal dominance is lost, resulting in failure of the method. The time-domain methods presented in this work alleviate both the efficient representation problem (by not relying on Fourier series expansions) and the dominance problem (the time-domain jacobian is diagonally dominant for most practical circuits).
Kundert et al (K. Kundert, J. White, and A. Sangiovanni-Vincentelli, "A Mixed Frequency-Time Approach for Distortion Analysis of Switching Filter Circuits," IEEE J. Solid-State Ckts., 24(2):443-451, April 1989) proposed a mixed frequency-time approach to simulate an important special case of the multi-tone problem, characterized by strong nonlinearity in only one tone and mild nonlinearity in the others. By representing signals in mildly nonlinear paths with Fourier series of few terms, they obtain a relation between two sets of time-domain samples, separated by the period of the strongly nonlinear tone. Another relation between these sets is obtained from the circuit's differential equations. Equating these two relations enables the mildly nonlinear tones of the quasi-periodic solution to be computed efficiently. Typically, the strongly nonlinear tone (e.g., clock) is much faster than the weakly nonlinear ones; this can lead to numerical ill-conditioning of this algorithm, because slowly-varying signals are extrapolated from their variation over one period of the fast tone.